We have analyzed the given file
and found that the file is corrupted due to an invalid named range. This can be
confirmed by opening the file in MS Excel itself. This can be repaired in MS
Excel, which results in removal of such named ranges. Whereas, XlsIO throws
exception and doesn’t repair it. This is not an issue. You can resave the file
with MS Excel and open in XlsIO, which resolves the issue.

Solved! I'm writing this both as a note to myself, and to anyone who may have the same problem in the future.

LibreOffice, prior to version 4.2, produced corrupted files when saving in MS Office XML-based formats (XLSX, DOCX, PPTX). Even when opening those files and saving them with a newer LibreOffice version, they still cannot be opened by MS Office programs (nor by XlsIO) without repairing.

I've found out a quick method for repairing LibreOffice-generated XLSX files using LibreOffice itself (tested with v4.3): just save the file in XLS format, close LibreOffice, then open the XLS file and save it in XLSX format again. No Excel required!

Of course it would still be nice if XlsIO had a way of just ignoring spurious data in XLSX files, but at least now there's a workaround of some sort.